include adding problem-specific cuts and employing customized node selection rules and branching priorities [81].
Since the computational complexity associated with the standard BnC method for solving the exact MISOCP reformulation may not be practically affordable in large-scale networks, we develop a low-complexity second-order cone programming (SOCP) based inflation pro-cedure (i.e., a greedy algorithm) [83, 84] to compute the near-optimal solutions of the RCBA problem. Different from the inflation procedures presented in Chapters 2 and 3, we invoke a sub-enumeration procedure (see Alg. 5.1) and solve a sequence of SOCPs to determine which one of the non-admitted MSs is the best candidate to admit in each iteration of the inflation procedure. The best candidate MS in each iteration is the one that results in the largest increase in the system utility if it is admitted. The inflation procedure proposed in this chapter represents a greedy algorithm.
The simulations results show that the MILP based approach, the MISOCP based ap-proach, and the inflation procedure yield almost the same average number of admitted MSs.
However, the MILP based approach requires much more total transmitted BS power to guar-antee the SINR targets of the admitted MSs than that of the MISOCP based approach and the inflation procedure. The numerical results also demonstrate that the inflation procedure has much less computational complexity than the MILP based approach and the MISOCP based approach when the number of admissible MSs is large. While the MISOCP based ap-proach achieves the largest system utility on average, it also admits the highest computational complexity among the three methods.
This chapter is based on my original work that has been published in [124, 125, 137], and the MILP based approach is added and new simulation results are presented in this chapter.
5.2 System model and problem statement
As in Chapters 3 and 4, in this chapter we focus on the downlink of a cellular network with one BS equipped with M transmit antennas, and K single-antenna MSs. The K MSs are admissible under the prescribed minimum received SINR targets (representing QoS require-ments [23]). As in Chapter 4, we denote hHk ∈ C1×M, uk ∈ CM×1, and pk > 0 as the frequency-flat channel vector, the unit-norm precoding vector, and the allocated transmis-sion power, respectively, of the kth MS, ∀k ∈ K , {1,2,· · ·, K}. The received signal
yk∈Cat thekth MS can then be written as (see, e.g., [12, 13, 18, 28]) yk=hHkuk√pkxk+
XK j=1,j6=k
hHkuj√pjxj +zk,∀k ∈ K (5.1)
wherexk ∈ Cdenotes the normalized data symbol, i.e.,E{|xk|2} = 1, intended to thekth MS, andzk ∈ C stands for the additive circularly-symmetric white Gaussian noise [19] at thekth MS, with zero mean and varianceσk2, ∀k ∈ K. Note that the signal model in (5.1) is identical to that given in Eq. (4.1) of Chapter 4. However, different from Chapter 4, it is assumed in this chapter that the downlink channel vectors{hk,∀k ∈ K}are notknown at the BS. That is, we assume in this chapter that the downlink channel vectors{hHk,∀k ∈ K}
are random vectors.
As in Chapter 4 (cf. Eq. (4.2)), we consider in this chapter the codebook-based multiuser downlink beamforming. That is, the normalized precoding vector uk is assigned from the predefined precoding vector codebookBconsisting ofL >1fixed precoding vectors, i.e.,
uk ∈ B,{v1,v2,· · · ,vL},∀k∈ K (5.2) where the precoding vectorvl ∈ CM×1 andkvlk2 = 1, ∀l ∈ L , {1,2,· · · , L}. Assume that the data symbols of the MSs are mutually independent and independent from the noise.
With single-user detection at the receivers, theaveragereceived SINR at thekth MS, denoted by SINRk, can then be expressed as (see, e.g., [12, 37–42])
SINRk , pkuHkRkuk
PK
j=1,j6=kpjuHj Rkuj+σk2,∀k ∈ K (5.3) where the matrix Rk , E
hkhHk ∈ CM×M represents the true CCM of the kth MS,
∀k ∈ K. Note that the term SINRk defined in (5.3) represents the average received SINR at the kth MS, while the expression SINRk given in Eq. (4.3) of Chapter 4 refers to the instantaneous received SINR at thekth MS [12].
Similar to Chapter 4, to model the precoding vector assignment procedure, we introduce thebinaryinteger variableak,l ∈ {0,1}to indicate withak,l = 1that thelth precoding vector vl ∈ B is assigned to the kth MS, and ak,l = 0otherwise. Accordingly, we introduce the continuous variableφk,l ≥ 0to model the transmission power allocated to thelth precoding vectorvl ∈ B for thekth MS,∀k ∈ K,∀l ∈ L. Since at most one precoding vector may be assigned to a MS in codebook-based downlink beamforming (also known as
single-layer-5.2. System model and problem statement 109 per-user precoding [7]), we impose the following constraints in the RCBA problem:
XK k=1
XL l=1
φk,l ≤P(MAX) (5.4)
0≤φk,l ≤ak,lP(MAX),∀k ∈ K,∀l∈ L (5.5) XL
l=1
ak,l ≤1,∀k∈ K (5.6)
where Eq. (5.4) represents the per-BS sum-power constraint, with the constantP(MAX) >0 denoting the maximum transmission power of the BS. Eq. (5.5) implements the so-called big-M method [67, 68] to ensure thatφk,l = 0whenak,l = 0. Furthermore, due to the per-BS sum-power constraint in (5.4), Eq. (5.5) is automatically satisfied whenak,l = 1. Note that for the precoding vector selection constraints, i.e., the multiple-choice constraints in (5.6), ifP
l=1ak,l = 0, i.e., if no precoding vector is assigned to thekth MS, thekth MS is not admitted in the current time-slot. Hence, with the multiple-choice constraints in (5.6), user admission control is naturally embedded in the precoding vector assignment procedure.
As in Chapter 4 (cf. Eqs. (4.9) and (4.10)), under the constraints in Eqs. (5.5) and (5.6), we can express the transmission powerpkand the beamformerukof thekth MS, respectively, as
pk = XL
l=1
ak,lφk,l = XL
l=1
φk,l,∀k ∈ K (5.7)
uk = XL
l=1
ak,lvl,∀k ∈ K. (5.8)
Eqs. (5.7) and (5.8) together further imply that (√pjuj)HRk(√pjuj) =
XL l=1
φj,lvlHRkvl = XL
l=1
φj,lTr
RkVl ,∀j, k ∈ K (5.9)
where the constant matrixVl ∈CM×M is defined as
Vl ,vlvHl 0,∀l ∈ L. (5.10)
Making use of Eq. (5.9), the term SINRkdefined in Eq. (5.3) can be rewritten as SINRk =
PL
m=1φk,mTr RkVm
PK j=1,j6=k
PL
l=1φj,lTr
RkVl +σk2,∀k ∈ K. (5.11) Due to limited channel training resources, channel variations, channel estimation errors, and channel feedback errors and delay, thetrueCCMRkis usually not available at the BS, e.g., in FDD systems [7,19]. In this case, only theestimatedCCM of thekth MS, denoted by
b
Rk ∈CM×M, is known to the BS [12,37–40]. In practical systems, the estimated CCMRbkis generally different from the true CCMRk. Following the approach presented in [12, 37–40], we model in this chapter the estimated (erroneous) CCMRbkas
Rbk =Rk+∆k,∀k ∈ K (5.12)
where the matrix ∆k ∈ CM×M denotes the estimation errors in the estimated CCM Rbk, i.e., the matrix∆krepresents the mismatch matrix. We know from practical considerations that the matrices Rk and Rbk are positive semidefinite, i.e., Rk 0 and Rbk 0, and the mismatch matrix∆k is Hermitian, i.e.,∆k =∆Hk. Further, it is commonly assumed in the literature that the Frobenius norm of the mismatch matrix∆kis upper-bounded by a known constantδk ≥0(see, e.g., [12, 37, 38]), i.e.,
k∆kkF ≤δk,∀k ∈ K. (5.13)
In this chapter, we consider the problem of precoding vector assignment and power al-location for theK MSs to maximize the system utility functionf({ak,l},{φk,l}), which is defined as
f({ak,l},{φk,l}), XK k=1
XL l=1
ak,l−ρ XK
k=1
XL l=1
φk,l (5.14)
where the constant weighting factor ρ > 0 is adopted to guarantee that maximizing the system utility functionf({ak,l},{φk,l})will result in the maximum number of admitted MSs (i.e., the termPK
k=1
PL
l=1ak,l) with the minimum total transmitted BS power (i.e., the term PK
k=1
PL
l=1φk,l) [83, 110, 111]. As in Chapter 3, taking into account the per-BS sum-power constraint in (5.4), we can simply choose the weighting factorρasρ= 1/ 1 +P(MAX)
[83, 110, 111].
Similarly as in the conventional QoS-constrained designs [12, 13, 18, 24–33, 37–42, 131–
133], if thekth MS is admitted, i.e., ifPL
l=1ak,l = 1, then the average received SINR of the
5.2. System model and problem statement 111 kth MS must exceed or equal to a prescribed thresholdΓ(MIN)k to guarantee the QoS that the kth MS is subscribed to. To achieve robustness against the CCM estimation errors{∆k,∀k ∈ K}, we adopt here the worst-case robust design approach (see, e.g., [12, 37, 38, 131–134]).
Specifically, we define in this chapter the following worst-case SINR constraints for theK admissible MSs:
∆mink∈Ek
SINRk= min
∆k∈Ek
PL
m=1φk,mTr bRk−∆k Vm PK
j=1,j6=k
PL
l=1φj,lTr bRk−∆k
Vl +σk2
≥Γ(MIN)k XL
l=1
ak,l,∀k ∈ K (5.15)
where the estimation error setEkis defined as Ek,n
∆k|Rk =Rbk−∆k0, and k∆kk2F ≤δk2o
,∀k ∈ K. (5.16) Note that the CCM estimation error sets {Ek,∀k ∈ K} defined in (5.16) are mutually independent among theKadmissible MSs.
With the system utility function f({ak,l},{φk,l}) defined in (5.14) and the worst-case SINR constraints defined in Eqs. (5.15) and (5.16), the robust joint codebook-based downlink beamforming and admission control (RCBA) problem can be stated as
Φ(RCBA) , max
{ak,l,φk,l} f({ak,l},{φk,l}) (5.17a) s.t. (5.4):
XK k=1
XL l=1
φk,l ≤P(MAX)
(5.5):0≤φk,l ≤ak,lP(MAX),∀k ∈ K,∀l ∈ L (5.6):
XL l=1
ak,l ≤1,∀k∈ K
(5.15) min
∆k∈Ek
PL
m=1φk,mTr bRk−∆k Vm PK
j=1,j6=k
PL
l=1φj,lTr bRk−∆k
Vl +σk2 ≥Γ(MIN)k XL
l=1
ak,l,∀k ∈ K (5.16):Ek ,n
∆k|Rk =Rbk−∆k 0, and k∆kk2F ≤δ2ko
,∀k ∈ K
ak,l ∈ {0,1},∀k∈ K,∀l∈ L. (5.17b)
The RCBA problem (5.17) contains the inner optimization problems in the worst-case SINR constraints in (5.15) and the outer optimization problem (5.17). As a result, the RCBA problem formulation in (5.17) represents a BL-MIP [60, 136], which is generally
intractable due to the inner optimization step in Eq. (5.15) and the integer constraints in Eq. (5.17b) [60, 136]. To facilitate the development of efficient algorithmic solutions, we de-rive a MILP approximation in next section and an exactly equivalent MISOCP reformulation in Section 5.4 of the RCBA problem in (5.17), respectively.